cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A180599 Zero followed by infinitely many 9's.

Original entry on oeis.org

0, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9
Offset: 0

Views

Author

Odimar Fabeny, Sep 10 2010

Keywords

Comments

Another interpretation: A real number with an infinitesimally small difference from the integer 1 which is used to test the precision of calculating devices. - John W. Nicholson, Feb 01 2012
a(n) is also the number of n-digit positive repdigit numbers (A010785). - Stefano Spezia, Aug 15 2020

Examples

			Viewed as a real number: For a TI-89, entering 1.-10^-12 yields .999999999999; however, 1.-10^-13 yields 1. - _John W. Nicholson_, Feb 01 2012

Links

Crossrefs

Cf. A007953, A010785, A010888, A057427, A180592, A180594, A180595, A180596, A180597, A180598.

Programs

Formula

a(0) = 0, a(n) = 9 for n > 0.
a(n) = 9 * A057427(n).
a(n) = A010888(9*n), where A010888 is the digital root.
From Robert Israel, Dec 16 2014: (Start)
G.f.: 9*x/(1 - x).
E.g.f.: 9*(exp(x) - 1). (End)

Extensions

More terms from Robert G. Wilson v, Sep 20 2010
Definition changed by N. J. A. Sloane, Feb 04 2012

A180592 Digital root of 2n.

Original entry on oeis.org

0, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 1
Offset: 0

Views

Author

Odimar Fabeny, Sep 10 2010

Keywords

Comments

Period 9. - Robert G. Wilson v, Sep 20 2010
Also digital root of A002276(n). - Enrique Pérez Herrero, Nov 05 2022

Links

Crossrefs

Cf. A007953, A010888, A180593, A180594, A180595, A180596, A180597, A180598, A180599.

Programs

Formula

From R. J. Mathar, Nov 02 2010: (Start)
a(n) = A010888(2*n).
a(n) = a(n-9), n > 9.
G.f.: -x*(2 + 4*x + 6*x^2 + 8*x^3 + x^4 + 3*x^5 + 5*x^6 + 7*x^7 + 9*x^8) / ( (x-1)*(1 + x + x^2)*(x^6 + x^3 + 1) ). (End)

Extensions

More terms from Robert G. Wilson v, Sep 20 2010
Keyword:base and formulas from R. J. Mathar, Nov 02 2010

A180597 Digital root of 7n.

Original entry on oeis.org

0, 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, 3, 1, 8, 6, 4, 2, 9, 7, 5, 3, 1, 8
Offset: 0

Views

Author

Odimar Fabeny, Sep 10 2010

Keywords

Comments

Period of 9. - Robert G. Wilson v, Sep 20 2010

Links

Crossrefs

Cf. A010888, A008589.
Cf. A180592, A180594, A180595, A180596, A180592, A180593, A180594, A180595, A180596, A180598, A180599.
Column k = 7 in A353109.

Programs

Formula

G.f.: x*(7 + 5*x + 3*x^2 + x^3 + 8*x^4 + 6*x^5 + 4*x^6 + 2*x^7 + 9*x^8)/(1 - x^9). - Stefano Spezia, Apr 21 2022
a(n) = A010888(A008589(n)). - Michel Marcus, Apr 21 2022

Extensions

More terms from Robert G. Wilson v, Sep 20 2010

A180593 Digital root of 3n.

Original entry on oeis.org

0, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6
Offset: 0

Views

Author

Odimar Fabeny, Sep 10 2010

Keywords

Comments

Decimal expansion of 41/1110. - Enrique Pérez Herrero, Nov 13 2021

Links

Crossrefs

Cf. A008585 (3*n), A010888 (digital root), A002277.
Cf. A180592, A180594, A180595, A180596, A180597, A180598, A180599.

Programs

  • Mathematica
    digitalRoot[n_Integer?Positive] := FixedPoint[Plus@@IntegerDigits[#]&,n]; Table[If[n==0,0,digitalRoot[3*n]], {n,0,200}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)
LinearRecurrence[{0,0,1},{0,3,6,9},120] (* Harvey P. Dale, Sep 03 2020 *)

Formula

a(n+1) = 3*A010882(n). - Reinhard Zumkeller, Oct 25 2010
G.f.: (-3*(1 + 2*x + 3*x^2))/(-1 + x^3) for n>0. - Alexander R. Povolotsky, Jun 13 2012
a(n) = A010888(A002277(n)). - Enrique Pérez Herrero, Nov 24 2022
a(n) = A010888(A008585(n)). - Michel Marcus, Nov 24 2022

Extensions

Edited by N. J. A. Sloane, Sep 23 2010

A180595 Digital root of 5n.

Original entry on oeis.org

0, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 1, 6, 2, 7
Offset: 0

Views

Author

Odimar Fabeny, Sep 10 2010

Keywords

Comments

Period of 9. - Robert G. Wilson v, Sep 20 2010

Links

Crossrefs

Cf. A007953, A010888, A180592, A180594.
Cf. A180592, A180593, A180594, A180596, A180597, A180598, A180599.

Programs

  • Mathematica
    f[n_] := Mod[5 n - 1, 9] + 1; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Sep 20 2010 *)
Join[{0}, ReplaceAll[Table[Mod[5n, 9], {n, 99}], {0 -> 9}]] (* Alonso del Arte, Sep 23 2012 *)
PadRight[{0},120,{9,5,1,6,2,7,3,8,4}] (* Harvey P. Dale, Jan 10 2024 *)

Formula

G.f.: -x*(9*x^8+4*x^7+8*x^6+3*x^5+7*x^4+2*x^3+6*x^2+x+5)/((x-1)*(x^2+x+1)*(x^6+x^3+1)). - Colin Barker, Sep 23 2012

Extensions

More terms from Robert G. Wilson v, Sep 20 2010

A180596 Digital root of 6n.

Original entry on oeis.org

0, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3, 9, 6, 3
Offset: 0

Views

Author

Odimar Fabeny, Sep 10 2010

Keywords

Comments

Period of 3. - Robert G. Wilson v, Sep 20 2010

Links

Crossrefs

Cf. A010888, A008588.
Cf. A180592, A180593, A180594, A180595, A180597, A180598, A180599.
Column k = 6 in A353109.

Programs

  • Mathematica
    f[n_] := Mod[6 n - 1, 9] + 1; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Sep 20 2010 *)
PadRight[{0},120,{9,6,3}] (* Harvey P. Dale, Dec 18 2012 *)

Formula

G.f.: 3*x*(2 + x + 3*x^2)/(1 - x^3). - Stefano Spezia, Apr 21 2022
a(n) = A010888(A008588(n)). - Michel Marcus, Apr 24 2022

Extensions

More terms from Robert G. Wilson v, Sep 20 2010

A353109 Array read by antidiagonals: A(n, k) is the digital root of n*k with n >= 0 and k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 1, 3, 3, 1, 6, 0, 0, 7, 3, 6, 7, 6, 3, 7, 0, 0, 8, 5, 9, 2, 2, 9, 5, 8, 0, 0, 9, 7, 3, 6, 7, 6, 3, 7, 9, 0, 0, 1, 9, 6, 1, 3, 3, 1, 6, 9, 1, 0, 0, 2, 2, 9, 5, 8, 9, 8, 5, 9, 2, 2, 0
Offset: 0

Views

Author

Stefano Spezia, Apr 24 2022

Keywords

Examples

			The array begins:
    0, 0, 0, 0, 0, 0, 0, 0, ...
    0, 1, 2, 3, 4, 5, 6, 7, ...
    0, 2, 4, 6, 8, 1, 3, 5, ...
    0, 3, 6, 9, 3, 6, 9, 3, ...
    0, 4, 8, 3, 7, 2, 6, 1, ...
    0, 5, 1, 6, 2, 7, 3, 8, ...
    0, 6, 3, 9, 6, 3, 9, 6, ...
    0, 7, 5, 3, 1, 8, 6, 4, ...
    ...

Crossrefs

Cf. A003991, A004247, A010888, A056992 (diagonal), A073636, A139413, A180592, A180593, A180594, A180595, A180596, A180597, A180598, A180599, A303296, A336225, A353128 (antidiagonal sums), A353933, A353974 (partial sum of the main diagonal).

Programs

  • Mathematica
    A[i_,j_]:=If[i*j==0,0,1+Mod[i*j-1,9]];Flatten[Table[A[n-k,k],{n,0,12},{k,0,n}]]
  • PARI
    T(n,k) = if (n && k, (n*k-1)%9+1, 0); \\ Michel Marcus, May 12 2022

Formula

A(n, k) = A010888(A004247(n, k)).
A(n, k) = A010888(A003991(n, k)) for n*k > 0.
Showing 1-7 of 7 results.

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